In this paper, we shall not concern ourselves
with compact objects like the white dwarf or a neutron star.
We shall assume that the matter is extremely massive, and such objects
continue to collapse under its own gravity.
The gravitational collapse phenomenon in GR show that the collapse outcome
depends upon, among other quantities, the choices of mass profiles and
velocity profiles of the collapsing matter. In the context of
inhomogeneous LTB models in GR,
these issues have been considered in great detail for various matter
models including dust and
viscous fluids [1]}.
The stellar collapse of stellar collapse in \(f(R)\)  gravity
using different forms of \(f(R)\)  function and different
matter distributions may be
found in [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, [14]}, [15]}, [16]}.
Although different type of \(f(R)\)  models may be considered, only the ones
which are in agreement with the standard cosmological observations
should be of interest. Here, we consider three
models of \(f(R)\)  gravity given respectively by: \((a)~f(R)=(R+\lambda R^{2})\)  [17]}  , 
\((b) ~f(R)\sim R^{1+\epsilon }\) , with \(\epsilon <<1\)  [7]} and \((c)~f(R)=R+\lambda \,\left[\exp {(-\sigma R)}-1\right]\) , with \(\lambda \sigma <1\) ,  [19]}.
All three theories play a major role in the study of the early and the late universe.
For example, in the \(R+\lambda R^2\)  model,
the \(R^{2}\)  term is weak in the weak gravity regime.
But it also contributes significantly during the early
universe and in strong gravity regime. Here, we
study gravitational collapse of an extremely massive object (so massive that its
gravitational attraction predominates thermal or quantum mechanical pressures), and so naturally,
since we are in a strong gravitational field, this
model may assume significance.
In all these theories, we shall show that an extremely
massive spherical matter cloud, admitting radial and tangential pressures, and outgoing
heat flux, can collapse
in a manner that no horizon is formed at the boundary (since the star
radiates off all its
mass before reaching the singularity) making the central singularity
naked but weakly singular. To ensure that the gravitational
collapse proceeds continuously without forming any stable object,
we assume the radius of the 2- sphere cross- sections decrease linearly with time.
The interior collapsing spacetime shall be smoothly matched with the
exterior
Vaidya spacetime [20]} over a timelike surface
\(\Sigma \)  [21]}.
